I am in the R.S. presidency and one family in our ward is now completely gone from my lds.org directory and my LDS Tools app. My R.S. counselor and my ward clerk both still see this family in both places, but I do not. Our R.S. president just visited this family, so I know they are still the ward. I'm wondering if there is something wrong with my account?

Most likely the privacy settings for the family has changed. If they've been set to "Leadership only" you won't see them, but the clerk can. Not sure about why your counselor can see it unless she hasn't synced her LDS Tools since the change.